On a conjecture about the second Zagreb index

Kinkar Chandra Das; Akbar Ali

Author(s)

Kinkar Chandra Das Department of Mathematics, Sungkyunkwan University,Suwon 16419,Republic of Korea

Akbar Ali Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea; Department of Mathematics, College of Science, University of Hail, Hail 81451, Saudi Arabia; Knowledge Unit of Science, University of Management and Technology, Sialkot 51310, Pakistan

Abstract

The cyclomatic number of a graph G is the minimum number of those edges of G whose removal makes G as acyclic. The sweictohn1d Zatgrebk ind1eaxnMd24 forka grnaph2G,leist GthebseutmheofgrtahpehphroadvuinctgsmofaxdiemgruemesMof2 avdajlauceenint tvheertciclaesssooffGa.ll Fcoonrnec=tedk g2r1ap+hst with order n and cyclomatic number . Xu et al. [MATCH Commun. Math. Comput. Chem. 72 (2014) 641-654] posed a conjecture concerning the exact structure of the graph G . In this note, a partial progress is made on this conjecture by proving that G has a specific type of subgraph with the size k 2 1 + t and minimum degree at least one.

Publication Details

Page(s) 38-43

DOI DOI not available

Published Journal: Discrete Mathematics Letters, Volume: 2, Issue: 0, Year: 2019

Request the full text from PASTIC

Keywords

extremal problem., cyclomatic number, second Zagreb index

References

Ali A. .2018 .Tetracyclic graphs with maximum second Zagreb index: a simple approach. , 11(5) : 1850064.

Ali A.,K. C. Das S.,Akhter S. . .On the extremal graphs for second Zagreb index with fixed number of vertices and cyclomatic number. Miskolc Math. Notes, : .

Bondy J. A.,Murty U. S. R. .2008 .. , : .

Borovic B.,K. C. Das B.,Furtula I. Gutman .2017 .Bounds for Zagreb indices. MATCH Commun. Math. Comput. Chem, 78 : 17-100.

Bollob B. .1999 .a´s, P. Erdo˝s, A. Sarkar, Extremal graphs for weights. , 200 : 5-19.

Deng H. .2007 .A unified approach to the extremal Zagreb indices for trees, unicyclic graphs and bicyclic graphs. MATCH Commun. Math. Comput. Chem, 57 : 597-616.

Gutman I.,Wilcox C. F. .1975 .Acyclic polyenes. J. Chem. Phys, 62 : 3399-3405.

Habibi N.,Zadeh T. D.,Ashrafi A. .2016 .Extremal tetracyclic graphs with respect to the first and second Zagreb indices. Trans. Combin, 5 : 35-55.

Harary F.,Addison-Wesley F. .1969 .. , : .

Xu K.,K. C. Das S.,Balachandran S. .2014 .Maximizing the Zagreb indices of (n; m)-graphs. MATCH Commun. Math. Comput. Chem, 72 : 641-654.